Helmut Koch Boeken

First published in German in 1970 and translated into Russian in 1973, this classic now becomes available in English. After introducing the theory of pro-p groups and their cohomology, it discusses presentations of the Galois groups G S of maximal p -extensions of number fields that are unramified outside a given set S of primes. It computes generators and relations as well as the cohomological dimension of some G S, and gives applications to infinite class field towers. The book demonstrates that the cohomology of groups is very useful for studying Galois theory of number fields; at the same time, it offers a down to earth introduction to the cohomological method. In a „Postscript“ Helmut Koch and Franz Lemmermeyer give a survey on the development of the field in the last 30 years. Also, a list of additional, recent references has been included.

Algebraic number theory is a sophisticated area of mathematics shaped by prominent mathematicians throughout history. This book aims to present the core elements of algebraic number theory, including normal extensions and an introduction to class field theory. It treats algebraic functions alongside algebraic numbers, highlighting the analogy between number fields and function fields, particularly when the ground field is finite. This approach also introduces 'higher congruences,' vital for 'arithmetic geometry.' Early chapters cover elementary number theory topics like Minkowski's geometry of numbers, public-key cryptography, and a concise proof of the Prime Number Theorem, inspired by Newman and Zagier. Subsequent chapters introduce essential tools such as ideals, discriminants, and valuations, applying these to function fields, including a proof of the Riemann-Roch Theorem and the first case of Fermat's Last Theorem via cyclotomic fields. The book also details Hecke $L$-series theory, following Tate, with applications to number theory, including the Generalized Riemann Hypothesis. Chapter 9 consolidates earlier material through quadratic number fields, while Chapter 10 offers an introduction to class field theory. The text aims to provide straightforward proofs, making it accessible for beginners eager to explore the depth of the subject. It is structured for two one-semester courses, with the first four chapters establishi

From the reviews: „... The author succeeded in an excellent way to describe the various points of view under which Class Field Theory can be seen. ... In any case the author succeeded to write a very readable book on these difficult themes.“ Monatshefte fuer Mathematik, 1994 „... Number theory is not easy and quite technical at several places, as the author is able to show in his technically good exposition. The amount of difficult material well exposed gives a survey of quite a lot of good solid classical number theory... Conclusion: for people not already familiar with this field this book is not so easy to read, but for the specialist in number theory this is a useful description of (classical) algebraic number theory.“ Medelingen van het wiskundig genootschap, 1995

Inhaltsverzeichnis: Einleitende Vorbemerkungen.- I. Problemstellung.- II. Aufbau der Arbeit.- III. Zusammenfassung der Ergebnisse.- Summary.- Erstes Kapitel: Die bisherige Wirtschaftsform.- 1.1 Die modernen Aufgabenstellungen der zivilen Luftfahrtindustrie im Verhältnis zu deren Leistungsvermögen.- Zweites Kapitel: Art und Umfang staatlicher Finanzhilfe.- 2.1 Vereinigte Staaten von Amerika.- 2.2 Frankreich.- 2.3 Großbritannien.- 2.4 Italien.- 2.5 Kanada.- 2.6 Schweiz.- 2.7 Südafrikanische Union.- 2.8 Japan.- 2.9 Bundesrepublik Deutschland.- 2.10 Europa.- 2.11 Zweites Zwischenergebnis.- Drittes Kapitel: Folgen für die Entscheidungsfreiheit der Privatwirtschaft.- Viertes Kapitel: Die Einflußnahme der Staaten auf die Privatwirtschaft zur Verbesserung ihrer Wettbewerbsfähigkeit.- 4.1 Größere Luftfahrtnationen.- 4.2 Kleinere Luftfahrtnationen.- 4.3 Europa.- 4.4 Bundesrepublik Deutschland.- 4.5 Drittes Zwischenergebnis: Zusammenfassung der Untersuchungen über die Einflußnahme der Staaten auf die Privatwirtschaft.- Fünftes Kapitel: Die Tätigkeit der Privatunternehmen unter Staatseinfluß.- 5.1 Vereinigte Staaten von Amerika.- 5.2 Großbritannien.- 5.3 Frankreich.- 5.4 Bundesrepublik Deutschland.- 5.5 Viertes Zwischenergebnis: Die Hauptkennzeichen der heutigen staatlich-privatwirtschaftlichen Kooperation in den verschiedenen Ländern und ihre Auswirkung auf dem Feld internationaler Konkurrenz.- Sechstes Kapitel: Die Verlagerung des Wettb